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seen Mar 24 at 17:09

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comment Explicit Bijection between Central Simple Algebras and twists of $\mathbb P^n$
@Bhargav: thanks for your addition. It's what I didn't want to write, because I wasn't as familiar with the descent argument. My bundle I and your bundle J are the same, right?
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answered Explicit Bijection between Central Simple Algebras and twists of $\mathbb P^n$
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revised Rationality of three-dimensional torus
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comment Rationality of three-dimensional torus
Basically, by taking character lattice of a torus, you are able to move back and forth between the world of algebraic tori and lattices of finite rank on which the Galois group acts. There is a theorem that says that a torus $T$ is stably rational if and only if the the character module $\hat{T}$ is stably permutation,, i.e. there exists a short exact sequence \begin{equation} 1 \rightarrow \hat{T} \rightarrow P_1 \rightarrow P_2 \rightarrow 1, \end{equation} with $P_1$, $P_2$ permutation modules. The book proves that such a sequence does not exist, so the torus is not stably rational.
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revised Rationality of three-dimensional torus
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answered Rationality of three-dimensional torus
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answered How is K-theory defined for coherent sheaves?
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